Nonabelian gerbes and branes

Urs Schreiber

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Aaron Bergman" <abergman@physics.utexas.edu> schrieb im Newsbeitrag\nnews:abergman-8630D7.23242304012005-100000@localhost...\n> In article\n> <Pine.LNX.4.31.0501040536380.24808-100000@feynman.harvard.edu>,\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n>\n>> On Mon, 3 Jan 2005, Aaron Bergman wrote:\n>>\n>> > In article <33soqpF43pvupU1-100000@individual.net>, Urs Schreiber\n>> > wrote:\n>>\n>> > > The gauge field on the D-brane is not part of this data but enters\n>> > > seperately.\n>> >\n>> > No. They\'re linked.\n>>\n>> Yes, they are linked, but the 1-form on the D-brane is not the 1-form in\n>> the gerbe cocycle.\n>\n> I\'ll have to read your paper. When I read JB\'s paper back when, I\n> thought it was clear that it was.\n\n\nLet me try to make this more precise, using the stuff from hep-th/0409200:\n\nAn F-string on a stack of D-branes in the presence of a Kalb-Ramond field is\nrelated to an abelian 1-gerbe G such that G restricted to the D-branes,\ndenoted G_Q with Deligne class\n\n[G_Q] = [lambda_ijk, alpha_ij, beta_i]_Q ,\n\nfulfills the relation\n\n[G_Q] = [1,0,B_Q] + [D(G_ij, A_i)] + [omega_ijk,0,0] .\n\nOn the left is the Deligne class of the abelian 1-gerbe, on the right is\n\n[1,0,B_Q] - the abelian gerbe coming from the Kalb-Ramond field B\nrestricted to Q where it can be taken to be globally defined\n\n[D(G_ij,A_i)] - the abelian lifting gerbe of the (possibly twisted)\nnonabelian bundle (G_ij, A_i) on the branes (D denotes a nonabelian\ngeneralization of the Deligne coboundary operator)\n\n[omega_ijk,0,0] - something related to spinors that I am going to ignore in\nthe following\n\nThis is how the gerbe, the Kalb-Ramond field and the D-brane gauge field A\nare related to each other. In particular, the 1-forms alpha_ij in the gerbe\nG cocycle (lambda_ijk, alpha_ij, beta_i) are not equal to the gauge field\n1-forms but related to them as\n\nalpha_ij = G_ij(d+A_j)G_ij^-1 - A_i\n\ni.e., they measure the twist in the connection on the branes.\n\nAs Aschieri and Jurco emphasize in their paper, the form of the right hand\nside essentially follows from a theorem that the gerbe on the left can be\nexpressed in terms of one of the form [1,0,B] with global B plus a lifting\ngerbe of a possibly twisted bundle. Hence this can be viewed as one way to\n*derive* the coupling of the boundary of the string to a nonabelian gauge\nfield from its coupling to the abelian KR 2-form.\n\nThis is of importance for the step to one dimension higher.\n\nA (twisted) nonabelian bundle can be called a (twisted) nonabelian 0-gerbe.\nHence we have here that a p-"brane" ending on a stack of branes is described\nby\n\n- an abelian p-gerbe coupled to the bulk\n\nand\n\n- a (twisted) nonabelian (p-1)-gerbe coupled to the boundary\n\nof the p-"brane", for p=1.\n\n\nGoing up one dimension the above scenario becomes concerned with a membrane\nending on a stack of 5-branes with a coupling to the abelian supergravity\n3-form C.\nC now plays a role analogous to B before.\n\nThe holonomy of the abelian 3-form C on a 3-cycle is computed by abelian\n2-gerbe holonomy. From results by Diaconescu, Moore and Freed\n(hep-th/0312069) it follows that this 2-gerbe is a "Chern-Simons 2-gerbe"\nwith respect to an E_8 Chern-Simons form. So this 2-gerbe class is denoted\n[CS.....] =: [CS].\n\nBy a reasoning completly analogous to that above, [CS] can be written as\n\n[CS] = [D G] + [1,0,0,C] + [theta_ijkl,0,0,0].\n\nHere D is again the nonabelian Deligne operator and G now denotes a possibly\ntwisted *nonabelian* 1-gerbe . D G is the abelian lifting 2-gerbe of that\npossibly twisted nonabelian 1-gerbe.\n\nBy comparison with the above, the nonabelian gerbe G with cocylce data G =\n(f_ijk, phi_ij, a_ij, d_ij, A_i, B_i) should describe the coupling of the\nboundary of the membrane to the 2-form B_i.\n\nThis 2-form is not the Kalb-Ramond 2-form but the nonabelian generalization\nof the abelian 2-forms found in the six-dimensional SCFT on the 5-brane\nworldvolume.\n\nSimilarly, the A_i enetering the nonabelian gerbe cocycle here are not in\nany obvious way related to D-brane gauge fields. They instead appear just as\nauxiliary as the transition functions phi_ij, for instance.\n\nSo once one understands holonomy of nonabelian 2-gerbes one should get that\n\na 2-brane ending on a stack of branes is described by\n\n- an abelian 2-gerbe coupled to the bulk\n\nand\n\n- a (twisted) nonabelian (2-1 = 1)-gerbe coupled to the boundary\n\nof the 2-brane,\n\njust as above for 1-dimension lower.\n\nThe closest relation of the nonabelian 1-gerbe cocylce data to a D-brane\ngauge field that I can see is that after compactifying the 5-branes on a\ncircle or torus the nonabelian B-field with one index in the compact\ndirection should give rise to the nonabelian gauge field on the resulting D4\nor D3 brane. This is at least what happen in the abelian case.\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Aaron Bergman" <abergman@physics.utexas.edu> schrieb im Newsbeitrag
news:abergman-8630D7.23242304012005-100000@localhost...
> In article
> <Pine.LNX.4.31.0501040536380.24808-1...an.harvard.edu>,
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>
>> On Mon, 3 Jan 2005, Aaron Bergman wrote:
>>
>> > In article <33soqpF43pvupU1-100000@individual.net>, Urs Schreiber
>> > wrote:
>>
>> > > The gauge field on the D-brane is not part of this data but enters
>> > > seperately.
>> >
>> > No. They're linked.
>>
>> Yes, they are linked, but the 1-form on the D-brane is not the 1-form in
>> the gerbe cocycle.
>
> I'll have to read your paper. When I read JB's paper back when, I
> thought it was clear that it was.

Let me try to make this more precise, using the stuff from http://www.arxiv.org/abs/hep-th/0409200:

An F-string on a stack of D-branes in the presence of a Kalb-Ramond field is
related to an abelian 1-gerbe G such that G restricted to the D-branes,
denoted [itex]G_Q[/itex] with Deligne class

[tex][G_Q] = [\lambda_ijk, \alpha_ij, \beta_i]_Q ,[/tex]

fulfills the relation

[tex][G_Q] = [1,0,B_Q] + [D(G_{ij}, A_i)] + [\omega_ijk,0,0] .[/tex]

On the left is the Deligne class of the abelian 1-gerbe, on the right is

[tex][1,0,B_Q] -[/itex] the abelian gerbe coming from the Kalb-Ramond field B
restricted to Q where it can be taken to be globally defined

[itex][D(G_{ij},A_i)] -[/itex] the abelian lifting gerbe of the (possibly twisted)
nonabelian bundle [itex](G_{ij}, A_i)[/itex] on the branes (D denotes a nonabelian
generalization of the Deligne coboundary operator)

[itex][\omega_ijk,0,0] -[/itex] something related to spinors that I am going to ignore in
the following

This is how the gerbe, the Kalb-Ramond field and the D-brane gauge field A
are related to each other. In particular, the 1-forms [itex]\alpha_ij[/itex] in the gerbe
G cocycle [itex](\lambda_ijk, \alpha_ij, \beta_i)[/itex] are not equal to the gauge field
1-forms but related to them as

[itex]\alpha_ij = G_{ij}(d+A_j)G_{ij}^-1 - A_i[/tex]

i.e., they measure the twist in the connection on the branes.

As Aschieri and Jurco emphasize in their paper, the form of the right hand
side essentially follows from a theorem that the gerbe on the left can be
expressed in terms of one of the form [1,0,B] with global B plus a lifting
gerbe of a possibly twisted bundle. Hence this can be viewed as one way to
*derive* the coupling of the boundary of the string to a nonabelian gauge
field from its coupling to the abelian KR 2-form.

This is of importance for the step to one dimension higher.

A (twisted) nonabelian bundle can be called a (twisted) nonabelian 0-gerbe.
Hence we have here that [itex]a p-[/itex]"brane" ending on a stack of branes is described
by

- an abelian p-gerbe coupled to the bulk

and

- a (twisted) nonabelian (p-1)-gerbe coupled to the boundary

of the p-"brane", for p=1.

Going up one dimension the above scenario becomes concerned with a membrane
ending on a stack of 5-branes with a coupling to the abelian supergravity
3-form C.
C now plays a role analogous to B before.

The holonomy of the abelian 3-form C on a 3-cycle is computed by abelian
2-gerbe holonomy. From results by Diaconescu, Moore and Freed
(http://www.arxiv.org/abs/hep-th/0312069) it follows that this 2-gerbe is a "Chern-Simons 2-gerbe"
with respect to an [itex]E_8[/itex] Chern-Simons form. So this 2-gerbe class is denoted
[itex][CS.....] =:[/itex] [CS].

By a reasoning completly analogous to that above, [CS] can be written as

[CS] [itex]= [D G] + [1,0,0,C] + [\theta_ijkl,0,0,0][/itex].

Here D is again the nonabelian Deligne operator and G now denotes a possibly
twisted *nonabelian* 1-gerbe . D G is the abelian lifting 2-gerbe of that
possibly twisted nonabelian 1-gerbe.

By comparison with the above, the nonabelian gerbe G with cocylce data G =
[itex](f_{ijk}, \phi_ij, a_{ij}, d_{ij}, A_i, B_i)[/itex] should describe the coupling of the
boundary of the membrane to the 2-form [itex]B_i[/itex].

This 2-form is not the Kalb-Ramond 2-form but the nonabelian generalization
of the abelian 2-forms found in the six-dimensional SCFT on the 5-brane
worldvolume.

Similarly, the [itex]A_i[/itex] enetering the nonabelian gerbe cocycle here are not in
any obvious way related to D-brane gauge fields. They instead appear just as
auxiliary as the transition functions [itex]\phi_ij,[/itex] for instance.

So once one understands holonomy of nonabelian 2-gerbes one should get that

a 2-brane ending on a stack of branes is described by

- an abelian 2-gerbe coupled to the bulk

and

- a (twisted) nonabelian [itex](2-1 =[/itex] 1)-gerbe coupled to the boundary

of the 2-brane,

just as above for 1-dimension lower.

The closest relation of the nonabelian 1-gerbe cocylce data to a D-brane
gauge field that I can see is that after compactifying the 5-branes on a
circle or torus the nonabelian B-field with one index in the compact
direction should give rise to the nonabelian gauge field on the resulting D4
or D3 brane. This is at least what happen in the abelian case.

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